"In those monarchies, most of the functions of the Queen are undertaken by a governor-general, but the Queen has a genuine discretion in appointing a governor-general, and also in responding to a request from a prime minister in one of these states to dismiss one. That, too, is a function that cannot be delegated. "
In Canada, at least, appointment of a governor general is only reserved for the Queen by convention. If she were to be indisposed when it came time for a new governor general to be appointed, the letters patent delegating powers to the governor general allow the old one to appoint the new one.
One of the delegated powers is "us[ing] Our Great Seal of Canada for sealing all things whatsoever
that may be passed under Our Great Seal of Canada." It also says that the governor general is appointed "by Commission under Our Great Seal of Canada." In any case, the letters patent are themselves sealed with the Great Seal of Canada, so they could be amended locally to allow the governor general to perform any function not delegated.
Dismissal is a little less certain, but it does allow the chief justice of the Supreme Court to exercise all powers in the event of the "incapacity" of the governor general, which is an expansive word and probably not reviewable in court.
"Indeed, the concept of regency does not apply to the other 15 monarchies. "
It does apply in New Zealand